Full text of "Treatise On Analysis Vol-Ii"

See other formats

```11    REGULARIZATION       295

If g is uniformly continuous with respect to a right-invariant distance on G,
the same argument applies, taking L — G.

(ii) There exists a function h e JTC(G) such that Np(g - h) g e (13.11.6).
By (14.10.6.1), it follows that NP((/B * g) - (fn * A)) g 0NP(# - A) g as. Thus
we are reduced to proving (ii) when g e Jf C(G). Let S be the support of g.
If V is a compact neighborhood of e, we have seen in (i) that fn * g - #
converges uniformly to 0 in the compact set K = S u VS. Next, if x \$ K,
we have

(/n * #)(*) - g(x) =

whence, by virtue of the Lebesgue-Fubini theorem and the invariance of
Haar measure

[ | (/„ *<?)(*) -<?(*) | 43(*)^ f  <#(*
JCK                                               Jo

= f   dft(s) [  |/B
J\>v           JG

Cv

• l^-1*)! dftx)

Since, on the other hand, the integral f  | (/„ * g)(x) — g(x) \ dfi(x) tends to 0

J K.

with l/«, we see that

and as N^^ * ^ — g) ^ (a + 1)11^11, the same is true of

N2(/n * g - g)2 g (a + 1)||0|| Nx(/n *g-g).

Suppose now that g e £"£ (G), and let /! e jSf c(G). If V is any compact
neighborhood of e in G, we have

^jn*9~gy^\ i-

Jv

f /,(j)#(j) f

Jv                Jo
```